A NOTE ON THE TURÁN FUNCTION OF EVEN CYCLES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) A NOTE ON THE TURÁN FUNCTION OF EVEN CYCLES OLEG PIKHURKO Abstract. The Turán function ex(n, F ) is the maximum number of edges in an F -free graph on n vertices. The question of estimating this function for F = C 2k, the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that ex(n, C 2k ) (k 1) n 1+1/k + 16(k 1)n, improving the previously best known general upper bound of Verstraëte [Combin. Probab. Computing 9 (2000), ] by a factor 8 + o(1) when n k. 1. Introduction The Turán function ex(n, F ) of a forbidden graph F is the maximum number of edges in an F -free graph on n vertices. It is named so as to honor the fundamental paper of Turán [20] from 1941 that determined this function for cliques. If we forbid C 2k, the cycle of length 2k, then the problem of determining its Turán function goes back even further, to a 1938 paper of Erdős [6] one of whose results is essentially that ex(n, C 4 ) = Θ(n 3/2 ). Determining the Turán function for even cycles is considered to be one of the key problems in extremal combinatorics. However, despite the efforts of many leading researchers, it remains wide open. In the proceedings of the 1963 Smolenice Symposium on Graph Theory and Its Applications, Erdős [7, Page 33] stated without proof that ex(n, C 2k ) γ k n 1+1/k for some constant γ k depending only on k. The first published proof of this appears in a paper of Bondy and Simonovits [3], whose Lemma 2 implies that ex(n, C 2k ) 20kn 1+1/k for all n k. More recently, Verstraëte [21], as a by-product of his theorem on cycle lengths in graphs, showed that ex(n, C 2k ) 8(k 1) n 1+1/k. The case that is best understood so far is k = 2. Here we know that ex(n, C 4 ) = (1/2 + o(1)) n 3/2 : the lower bound was proved by Erdős and Rényi [8] (and independently re-discovered by Brown [4]) and the upper bound by Erdős, Rényi, and Sós [9]. Moreover, we know ex(n, C 4 ) exactly when n = q 2 + q + 1 for any prime power q 16 (Füredi [11, 12]) and when n 31 (McCuaig [17], Clapham, Flockhart, and Sheehan [5], Yuansheng and Rowlinson [23]). For arbitrary fixed k, Erdős and Simonovits [10] conjectured that ex(n, C 2k ) = (1/2 + o(1)) n 1+1/k. This was disproved by Lazebnik, Ustimenko, and Woldar [15] for k = 5 who showed that ex(n, C 10 ) (4 5 6/5 + o(1)) n 6/5 = ( o(1)) n 6/5, 2010 Mathematics Subject Classification. Primary 05C35. Partially supported by the National Science Foundation, Grant DMS c XXXX American Mathematical Society

2 2 OLEG PIKHURKO although they were not aware of this conjecture at the time, see the discussion in [16, Pages ]. Füredi, Naor, and Verstraëte [13] showed that n 4/3 ex(n, C 6 ) n 4/3, thus disproving the case k = 3 of the above conjecture. For k 3, a lower bound of the form ex(n, C 2k ) = Ω(n 1+1/k ) is known only for k = 3 and k = 5 (first proved by Benson [1]). Some other constructions achieving it were found by Wenger [22], Lazebnik and Ustimenko [14], Mellinger [18], and Mellinger and Mubayi [19]. Unfortunately, for no other k is the rate of growth of ex(n, C 2k ) known, although there are various lower bounds. We refer the reader to the paper [16] that presents new constructions as well as gives numerous references. In this note we prove the following result that improves the best known general upper bound of Verstraëte [21] by a factor 8 + o(1) when n k. Theorem 1.1. For all k 2 and n 1, we have ex(n, C 2k ) (k 1) n 1+1/k + 16(k 1)n. Essentially all the main ideas that we use to prove this result can be found in the papers [3, 21]. However, given the importance of this problem and the absence of any improvements in upper bounds on ex(n, C 2k ) for general k in the last decade, this result may help to draw more interest to this area and to introduce some beautiful ideas from [3, 21] to a larger audience, especially that our proof seems to be simpler and more transparent than those in [3, 21]. Although the bound of Theorem 1.1 can be somewhat improved (especially for small k), the author could not show that lim inf k lim inf n ex(n, C 2k ) kn 1+1/k is strictly below 1. It would be very interesting to decide if the limit inferior is 0 or not. Hopefully, the quest in this direction will lead to new ideas and insights. 2. Auxiliary Results We use the standard graph theory notation that can be found in e.g. Bondy and Murty s book [2]. Still, some terms are defined when they are used for the first time. By a Θ-graph we mean a cycle of length at least 2k with a chord. We will need the following lemma that appears implicitly in the proof of Lemma 2 in [3] and is stated as a separate lemma in [21, Lemma 2]. Lemma 2.1. Let F be a Θ-graph and 1 l v(f ) 1. Let V (F ) = A B be an arbitrary partition of its vertex set into two non-empty parts such that every path in F of length l that begins in A necessarily ends in A. Then F is bipartite with parts A and B. Sketch of Proof. Let n = v(f ) and let its vertex set be Z n, the set of the residues modulo n, with i being adjacent to i 1 and i + 1. We encode the partition A B by a 2-coloring χ of Z n. Let P = {m Z n : i Z n χ(i) = χ(i + m)}

3 TURÁN FUNCTION OF EVEN CYCLES 3 consist of all periods of χ. Our assumption implies that l P. The smallest positive period m P has to divide n and, in fact, P = {mi : i Z n } is the set of multiples of m. Assume that m > 2 for otherwise we are easily done. Let the chord connect 0 to r. Since m > 2, we cannot have that both r and n r are congruent to 1 modulo m, say r 1 (mod m). Thus r 1 P. By the m-periodicity of χ, there is some j Z n such that χ(j) χ(j + l + r 1) and we can further assume that m < j 0. The l-walk j, j + 1,..., 1, 0, r, r + 1,..., j + l + r 1 in F connects A to B, which is a contradiction unless l + r 1 n. The remaining cases can be settled by similar arguments where the constructed l-path may change direction at a chord s endpoint. A complete proof can be found in [21, Lemma 2]. The following easy lemma will also be used. Lemma 2.2. Let k 3. contains a Θ-subgraph. Any bipartite graph H of minimum degree at least k Proof. Take a longest path P in H. Let P visit vertices x 1,..., x m in this order. The end-point x 1 has at least k neighbors in H. By the maximality of P, all of them lie on P. So pick any k neighbors x i1,..., x ik of x 1 where i 1 < < i k. Every two neighbors of x 1 are at least 2 apart on P (because H is bipartite). Thus i k 2k and the subpath of P between x 1 and x ik together with the edges x 1 x i2 and x 1 x ik forms the required Θ-subgraph. 3. Proof of Theorem 1.1 Suppose for the sake of contradiction that some C 2k -free graph G on n vertices violates Theorem 1.1. As is well known (for a proof see e.g. [3, Page 99] or [2, Theorem 2.5]), every graph G contains a subgraph of minimum degree at least half of the average degree of G: (1) G H G δ(h) d(g) = e(g) 2 n. So take any H G of minimum degree at least δ, where we define δ = e(g) n (k 1) n1/k + 16(k 1). Fix an arbitrary vertex x in H. Let V i consist of those vertices of H that are at distance i (with respect to the graph H) from x. Thus V 0 = {x} and V 1 = N(x) is the neighborhood of x in H. For i 0, let v i = V i and let H i = H[V i, V i+1 ] be the bipartite subgraph of H induced by the disjoint sets V i and V i+1. Claim 3.1. For 1 i k 1, neither of the graphs H[V i ] and H i contains a Θ-subgraph that is bipartite. Proof of Claim 3.1. Suppose on the contrary that a bipartite Θ-subgraph F H[V i ] exists. Let Y Z be the bipartition of F. Let T H be a breadth-first search tree in H with the root x (for definitions see e.g. [2, Section 6.1]). Let y be the vertex that is farthest from x such that every vertex of Y is a T -descendant of y. The paths inside T that connect y to Y branch at y. Pick one such branch,

4 4 OLEG PIKHURKO defined by some child z of y, and let A be the set of the T -descendants of z that lie in Y. Let B = (Y Z) \ A. Since Y \ A, B is not an independent set of F. Let l be the distance between x and y. We have l < i and 2k 2i + 2l < 2k v(f ). By Lemma 2.1 we can find a path P F of length 2k 2i + 2l that starts in some a A and ends in b B. Since the length of P is even, we have b Y. Let P a and P b be the unique paths in T that connect y to respectively a and b. They intersect only in the vertex y since P a starts with the edge yz while P b uses some different child of y. Also, each of these paths has length i l. But then the union of the paths P, P a, and P b forms a 2k-cycle in H, a contradiction. The same proof (where we let Y = V (F ) V i ) works for H i = H[V i, V i+1 ]. By (1), Lemma 2.2, and Claim 1 (and the simple fact that every graph has a bipartite subgraph with at least half the edges, see e.g. [2, Theorem 2.4]), we conclude that for k 3 we have (2) d(h[v i ]) 4k 4 and d(h i ) 2k 2, for 1 i k 1. Also, if k = 2, then H[V 1 ] has no path of length 2 and no vertex of V 2 can send more than one edge to V 1, so (2) still holds. We are essentially done with the combinatorial part and what remains is some algebra. Here is a sketch for n k. For i k 1, the inequalities in (2) imply that, on average, a vertex of V i sends at least δ O(k) edges to V i+1 while at most 2k 2 edges per vertex of V i+1 are sent back. Thus the first k ratios v i+1 /v i are at least (δ O(k))/(2k 2) each. We conclude that (δ/(2k 2)) k n + o(n), giving e(g) (2k 2 + o(1)) n 1+1/k. An extra factor of 2 + o(1) is saved by observing that, in view of v i v i+1, a vertex of V i+1 sends only k 1 + o(1) edges back on average. Let us provide all detailed calculations. Define ε = 4(k 1) 2 /δ. Let us show inductively on i = 0, 1,..., k 1 that (3) e(h i ) (k 1 + ε) v i+1, which bounds the average degree of the vertices in V i+1 into V i. Clearly, this is true for i = 0 since each vertex of V 1 sends only one edge to V 0. Suppose that we want to prove (3) for some i > 0. By (2) and the inductive assumption, (4) e(h i ) = y V i d Vi+1 (y) ( δ (4k 4) (k 1 + ε) ) v i = (δ 5k + 5 ε) v i. Thus the average degree of the vertices of V i with respect to H i is at least δ 5k + 5 ε 2k 2. Here we used the facts that (5) δ 16(k 1) and ε k 1. 4 In particular, V i+1. In order to satisfy the second inequality in (2), it must be the case that the average V i -degree of a vertex in V i+1 is at most 2k 2, that is, e(h i ) (2k 2)v i+1. Thus, by (4), we have v i By (2) we conclude that e(h i ) δ 5k + 5 ε 2k 2 δ 5k + 5 ε v i+1. 2e(H i ) (1 + 2k 2 δ 5k+5 ε ) v 2e(H i) = d(h i ) 2k 2. i+1 v i+1 + v i

5 TURÁN FUNCTION OF EVEN CYCLES 5 which implies the desired bound (3). (Indeed, (5) implies that 2(k 1) 2 δ 5.25(k 1) 4(k 1)2 δ 2(k 1) which, again by (5), gives 2 δ 5k+5 ε ε and the last inequality is exactly what we need for deducing (3).) By (3) and (4), we conclude that for each i = 0,..., k 1 we have v i+1 = v i+1 v i e(h i ) e(h i) δ 5k + 5 ε δ v i k 1 + ε k 1 + 4ε, where the last inequality follows again from (5). Since δ (k 1) n 1/k, we have ( ) k ( ) k δ e(g)/n n v(h) v k, k 1 + 4ε k (k 1) n 1/k implying the desired upper bound on e(g) (that is, a contradiction). This finishes the proof of Theorem 1.1. Acknowledgments The author thanks David Conlon and the anonymous referee for helpful remarks. References 1. C. T. Benson, Minimal regular graphs of girths eight and twelve, Can. J. Math. 26 (1966), J. A. Bondy and U. S. R. Murty, Graph theory, Springer Verlag, J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. Theory (B) 16 (1974), W. G. Brown, On graphs that do not contain a Thomsen graph, Can. Math. Bull. 9 (1966), C. R. J. Clapham, A. Flockhart, and J. Sheehan, Graphs without four-cycles, J. Graph Theory 13 (1989), P. Erdős, On sequences of integers no one of which divides the product of two others and on some related problems., Inst. Math. Mech. Univ. Tomsk 2 (1938), P. Erdős, Extremal problems in graph theory, Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Publ. House Czechoslovak Acad. Sci., Prague, 1964, pp P. Erdős and A. Rényi, On a problem in the theory of graphs, Publ. Math. Inst. Hung. Acad. Sci. 7 (1962), , In Hungarian. 9. P. Erdős, A. Rényi, and V. T. Sós, On a problem of graph theory, Stud. Sci. Math. Hungar. 1 (1966), P. Erdős and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), Z. Füredi, Graphs without quadrilaterals, J. Combin. Theory (A) 34 (1983), Z. Füredi, On the number of edges of quadrilateral-free graphs, J. Combin. Theory (B) 68 (1996), Z. Füredi, A. Naor, and J. Verstraëte, On the Turán number for the hexagon, Adv. Math. 203 (2006), F. Lazebnik and V. A. Ustimenko, Explicit construction of graphs with an arbitrary large girth and of large size, Discrete Applied Math. 60 (1995), , ARIDAM VI and VII (New Brunswick, NJ, 1991/1992). 15. F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, Properties of certain families of 2k-cycle-free graphs, J. Combin. Theory (B) 60 (1994), F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, Polarities and 2k-cycle-free graphs, Discrete Math. 197/198 (1999), , 16th British Combinatorial Conference (London, 1997). 17. W. McCuaig, 1985, unpublished letter, see [12, Page 1]. 18. K. E. Mellinger, LDPC codes from triangle-free line sets, Des. Codes Cryptogr. 32 (2004), K. E. Mellinger and D. Mubayi, Constructions of bipartite graphs from finite geometries, J. Graph Theory 49 (2005), 1 10.

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